# IGCSE Mathematics Foundation: Polygons

## Scheme of work: IGCSE Foundation: Year 10: Term 5: Polygons

#### Prerequisite Knowledge

1. Calculate Angles in Triangles
• Concept: Understanding that the sum of the interior angles in a triangle is always 180 degrees.
• Example: $\text{In a triangle with angles } 50^\circ \text{ and } 60^\circ, \text{ find the third angle.}$ $50^\circ + 60^\circ + x = 180^\circ.$ $x = 180^\circ – 110^\circ = 70^\circ.$
2. Calculate Angles in Parallel Lines
• Concept: Understanding the properties of alternate, corresponding, and co-interior angles formed by a transversal intersecting parallel lines.
• Example: $\text{If two parallel lines are intersected by a transversal and one of the alternate angles is } 120^\circ, \text{ find the other alternate angle.}$ $\text{The other alternate angle is also } 120^\circ.$ $\text{If one of the corresponding angles is } 120^\circ, \text{ the corresponding angle is also } 120^\circ.$

#### Success Criteria

1. Recognise and Use the Angle Properties of Polygons
• Objective: Identify and apply the angle properties specific to different types of polygons, such as triangles, quadrilaterals, pentagons, etc.
• Example: $\text{Identify the internal angles of a regular pentagon. Each internal angle of a regular pentagon is } \frac{(n-2) \times 180^\circ}{n} = \frac{3 \times 180^\circ}{5} = 108^\circ.$
2. Calculate and Use the Sums of the Interior Angles of Polygons
• Objective: Calculate the sum of the interior angles of any polygon using the formula $$(n-2) \times 180^\circ$$, where $$n$$ is the number of sides.
• Example: $\text{Calculate the sum of the interior angles of a hexagon.}$ $\text{Sum of interior angles} = (6-2) \times 180^\circ = 4 \times 180^\circ = 720^\circ.$
3. Calculate and Use the Sums of the Exterior Angles of Polygons
• Objective: Understand that the sum of the exterior angles of any polygon is always $$360^\circ$$, and use this property to find unknown exterior or interior angles.
• Example: $\text{Calculate the exterior angle of a regular octagon.}$ $\text{Each exterior angle} = \frac{360^\circ}{n} = \frac{360^\circ}{8} = 45^\circ.$

#### Key Concepts

1. Recognise and Use the Angle Properties of Polygons
• Concept: Polygons have specific angle properties based on their number of sides. Regular polygons have equal interior and exterior angles.
2. Calculate and Use the Sums of the Interior Angles of Polygons
• Concept: The sum of the interior angles of a polygon can be determined using the formula $$(n-2) \times 180^\circ$$, where $$n$$ is the number of sides. This formula is derived from the fact that a polygon can be divided into $$(n-2)$$ triangles.
3. Calculate and Use the Sums of the Exterior Angles of Polygons
• Concept: The sum of the exterior angles of any polygon is always $$360^\circ$$. This property holds true regardless of the number of sides in the polygon.

#### Common Misconceptions

1. Recognise and Use the Angle Properties of Polygons
• Common Mistake: Confusing the properties of regular and irregular polygons, particularly in identifying equal angles in regular polygons.
• Example: $\text{Incorrect: Assuming all polygons with the same number of sides have equal angles, even if they are not regular.}$ $\text{Correct: Recognize that only regular polygons have equal angles.}$
2. Calculate and Use the Sums of the Interior Angles of Polygons
• Common Mistake: Incorrectly using the formula for the sum of the interior angles, particularly miscounting the number of sides or triangles.
• Example: $\text{Incorrect: Calculating the sum of the interior angles of a pentagon as } 5 \times 180^\circ = 900^\circ.$ $\text{Correct: Using the formula } (n-2) \times 180^\circ = 3 \times 180^\circ = 540^\circ \text{ for a pentagon.}$
3. Calculate and Use the Sums of the Exterior Angles of Polygons
• Common Mistake: Misunderstanding that the sum of the exterior angles is always $$360^\circ$$, regardless of the number of sides.
• Example: $\text{Incorrect: Assuming the sum of the exterior angles of a hexagon is greater than that of a triangle because a hexagon has more sides.}$ $\text{Correct: Knowing that the sum of the exterior angles is always } 360^\circ \text{ for any polygon.}$

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